A Common Fixed Point Theorems in Menger Space using Occationally Weakly Compatible Mappings Kamal Wadhwa, Jyoti Panthi and Ved Prakash Bhardwaj Govt. Narmada Mahavidyalaya, Hoshangabad, (M.P) India Abstract In this paper we have improved the result of Saurabh Manro [7] by using the concept of occasionally weakly compatible Maps and proved some results on fixed points in menger space. Key words: Menger space, Common fixed point, occasionally weakly compatible mappings. 1. Introduction: In 1942 Menger [4] introduced the notion of a probabilistic metric space (PM-space) which is in fact, a generalization of metric space. The idea in probabilistic metric space is to associate a distribution function with a point pair, say (x, y), denoted by F(x, y; t) where t > 0 and interpret this function as the probability that distance between x and y is less than t, whereas in the metric space the distance function is a single positive number. Sehgal [8] initiated the study of fixed points in probabilistic metric spaces. The study of these spaces was expanded rapidly with the pioneering works of Schweizer-Sklar [1]. A weakly compatible map in fuzzy metric space is generalized by A. Al. Thagafi and Nasser Shahzad [1] by introducing the concept of occasionally weakly compatible mappings. Our paper improves the result of Saurabh Manro [7] by using of occasionally weakly compatible Maps and proved some results on fixed points in menger space. 2. Preliminaries: First, recall that a real valued function f defined on the set of real numbers is known as a distribution function if it is nondecreasing, continuous and inf f(x) = 0, sup f(x) = 1. We will denote by L, the set of all distribution functions. Definition 2.1: A probabilistic metric space (PM-space) is a pair (X, F) where X is a set and F is a function defined on X X to L such that if x, y and z are points of X, then (F-1), = 1 for every t > 0 iff x = y, (F-2), 0 = 0, (F-3), =,, (F-4) if, = 1 and, = 1, then, + = 1 for all x, y, z X and s, t 0. For each x, y X and for each real number t > 0,, is to be thought of as the probability that the distance between and y is less than t. It is interesting to note that, if (X, d) is a metric space, then the distribution function F(x, y; t) defined by the relation F(x, y; t) defined by the relation F(x, y; t) = H(t d(x, y)) induces a PM-space where H(x) denotes the distribution function defined as follows: H(x) = 0 0 1 > 0 Definition 2.2: A t-norm is a 2-place function, t:[0,1] [0,1] [0,1] satisfying the following: (i) t(0,0) = 0, (ii) t(0,1) = 1, (iii) t(a, b) = t(b, a), (iv) if a c, b d, then t(a, b) t(c, d), (v) t(t(a, b),c) = t(a, t(b, c)) for all a, b, c [0,1]. Definition 2.3: A Menger PM-space is a triplet (X, F, t) where (X, F) is a PM-space and t is a t-norm with the following condition: (F-5) F x, z (s + p) t(f x, y (s), F y, z (p)),for all x, y, z X and s, p 0. This inequality is known as Menger s triangle inequality. In our theory, we consider (X, F, t) to be a Menger PM-space with the additional following postulate: (F-6), t = 1 x, y X. Definition 2.4: A menger space (X, F, t) is said to be complete if and only if every Cauchy sequence in X is convergent. In 1996, Jungck [2] introduced the notion of weakly compatible maps as follows: Definitoin 2.5: A pair of self mappings (A, S) on set X is said to be weakly compatible if they commute at the coincidence points i.e. Au = Su for some u X, then SAu = ASu. We need the following Lemmas due to Schweizer and Skalr [1] and Singh and Pant [6], in the proof of the theorems: 39
Lemma 2.1: Let (X, F, t) be a menger space and if for a number k (0,1) such that,,. Then x = y. Definition: Let X be a set, f and g selfmaps of X. A point x X is called a coincidence point of f and g iff fx = gx. We shall call w = fx = gx a point of coincidence of f and g. Definition 2.6[3]: Two self mappings A and S of a non-empty set X are OWC iff there is a point x X which is a coincidence point of A and S at which A and S commute. The notion of OWC is more general than weak compatibility (see [5]). Lemma 2.2[3]: Let X be a non-empty set, A and B are occasionally weakly compatible self maps of X. If A and B have a unique point of coincidence, w = Ax = Bx, then w is the unique common fixed point of A and B. 3. Main Results: In our result, we used the following implicit relation: Definition (Implicit Relation): Let I= [0, 1] and Ω be the set of all real continuous functions φ : I 6 R satisfying the condition: (i) φ is non increasing or non decreasing in third and fourth argument and (ii) If we have φ(u, v, 1, 1, v, v) 1, for all u, v (0, 1) u v. Example: We define φ : I 6 R by φ(u 1, v 1, v 2, v 3, v 4, v 5 ) = u 1 - v 1 + v 2 - v 3 + v 4 - v 5 Then clearly continuous function such that if we have φ(u, v, 1, 1, v, v) 1, for all u, v (0, 1), Then φ(u, v, 1, 1, v, v) = u - v+ 1-1+ v- v = u v 1 u v. Theorem 3.1: Let (X,F,t) be a Menger space. Let A, B, S and T be self maps of X satisfying the following conditions: 1. (A, S) and (B, T) are owc. 2. there exist k (0,1) and such that! ",# $,! %,& $,! ",% $,! #,& $,! ",& $,! #,% $' 1 (I) for all x, y X and t > 0. Then there exists a unique point w X such that Aw = Sw = w and a unique point z X such that Bz = Tz = z. Moreover, z = w, so that there is a unique common fixed point A, B, S and T in X. Proof: Since the pairs (A, S) and (B, T) are owc, there exist points x, y X such that Ax = Sx, ASx = SAx and By = Ty, BTy = TBy. Now we show that Ax = By. Then we have by inequality (I),! ",# $,! %,& $,! ",% $,! #,& $,! ",& $,! #,% $' 1! ",# $,! %,& $,! ",% $,! #,& $,! ",& $,! #,% $' 1! ",# $,! ",# $,! "," $,! #,# $,! ",# $,! #," $' 1! ",# $,! ",# $,1,1,! ",# $,! #," $' 1! ",# $'! ",# $ Thus by lemma 2.1 Ax = By. Therefore Ax = Sx= By = Ty. Moreover, if there is another point z such that Az = Sz. Then using inequality (I) it follows that Az = Sz = By = Ty, or Ax = Az. Hence w = Ax = Sx is the unique point of coincidence of A and S. By lemma 2.2, w is the unique common fixed point of A and S. Similarly, there is a unique point z X such that z = Bz = Tz. Suppose that w z and using inequality (I), we get! (, $,! (, $,! (,( $,!, $,! (, $,!,( $' 1! (, $,! (, $,1,1,! (, $,!,( $' 1! (, $'! (, $ Thus by lemma 2.1 w = z. Therefore z = Sz = Tz = Az = Bz. To prove uniqueness, let u and v are two common fixed points of A, B, S and T in X. Therefore, by definition, Au = Bu = Tu = Su = u and Av = Bv = Tv = Sv = v. Then by (I), take x = u and y = v, we get! ),* $,! ),* $,! ),) $,! *,* $,! ),* $,! *,) $' 1! ),* $,! ),* $,1,1,! ),* $,! *,) $' 1 40
! ),* $'! ),* $ Therefore, by lemma 2.1, u = v. Hence the self maps A, B, S and T have a unique common fixed point in X. Theorem 3.2: Let (X, M, t) be a menger space and let A, B, S, T, P and Q be self maps of X satisfying the following conditions: (3.2.1) the pairs (A, SP) and (B, TQ) are owc; (3.2.2) there exists k (0,1) and φ Ω such that φ+f -,. kt,f 01,23 t,f -,01 t,f.,23 t,f -,23 t,f.,01 t4 1, x, y X and t > 0, (3.1.3) the pairs (A, P), (S, P), (B, Q) and (T, Q) are commuting; then A, B, S, T, P and Q have a unique common fixed point in X. Proof: Since the pairs (A, SP) and (B, TQ) are owc, so there are points x, y X such that Ax = SPx implies A(SP)x = (SP)Ax and By = TQy implies B(TQ)y = (TQ)By. We claim that Ax = By. Now by inequality (3.2.2) φ+f -,. kt,f 01,23 t,f -,01 t,f.,23 t,f -,23 t,f.,01 t4 1, We have, φ+f -,. kt,f -,. t,f -,- t,f.,. t,f -,. t,f.,- t4 1, φ+f -,. kt,f -,. t,1,1,f -,. t,f.,- t4 1, F -,. kt F -,. t, thus by lemma 2.1 Ax=By. Therefore Ax = SPx = By = TQy = z (say), then Az = SPz and Bz = TQz. We claim that Az = Bz. Now by inequality (3.2.2) φ+f -,. kt,f 01,23 t,f -,01 t,f.,23 t,f -,23 t,f.,01 t4 1, We have, φ+f -5,.5 kt,f 015,235 t,f -5,015 t,f.5,235 t,f -5,235 t,f.5,015 t4 1, φ+f -5,.5 kt,f -5,.5 t,1,1,f -5,.5 t,f.5,-5 t4 1, F -5,.5 kt F -5,.5 t, thus by lemma 2.1 Az = Bz. Therefore Az = SPz = Bz = TQz. Now we prove Az = z, Now by inequality (3.2.2), we have (by taking x = z and By = z) φ+f -5,5 kt,f -5,5 t,f -5,-5 t,f 5,5 t,f -5,5 t,f 5,-5 t4 1, φ+f -5,5 kt,f -5,5 t,1,1,f -5,5 t,f 5,-5 t4 1, F -5,5 kt F -5,5 t, thus by lemma 2.1 Az = z. Therefore z = Az = Bz = SPz = TQz. Now we put x = Pz and y = z in inequality (3.2.2) we get φ+f -15,.5 kt,f 0115,235 t,f -15,0115 t,f.5,235 t,f -15,235 t,f.5,0115 t4 1, Since (A, P) and (S, P) are commuting; φ+f 15,5 kt,f 15,5 t,f 15,15 t,f 5,5 t,f 15,5 t,f 5,15 t4 1, φ+f 15,5 kt,f 15,5 t,1,1,f 15,5 t,f 5,15 t4 1, F 15,5 kt F 15,5 t, thus by lemma 2.1 Pz = z. Since z = SPz Sz = z. To show Qz = z, we put x = z and y = Qz in inequality (3.2.2) we get φ+f -5,.35 kt,f 015,2335 t,f -5,015 t,f.35,2335 t,f -5,2335 t,f.35,015 t4 1, Since (B, Q) and (T, Q) are commuting; φ+f 5,35 kt,f 5,35 t,f 5,5 t,f 35,35 t,f 5,35 t,f 35,5 t4 1, φ+f 5,35 kt,f 5,35 t,1,1,f 5,35 t,f 35,5 t4 1, F 35,5 kt F 35,5 t, thus by lemma 2.1 Qz = z. Since z = TQz Tz = z. Therefore Az = Bz = Sz = Tz = Pz = Qz = z. i.e. z is the common fixed point of A, B, S, T, P and Q. To prove uniqueness: let r and s be two distinct common fixed points of A, B, S, T, P and Q. Then Ar = Br = Sr = Tr = Pr = Qr = r and As = Bs = Ss = Ts = Ps = Qs = s, Now by inequality (3.1.2), we have (at x = r and y = s) φ+f 6,7 kt,f 6,7 t,f 6,6 t,f 7,7 t,f 6,7 t,f 7,6 t4 1, φ+f 6,7 kt,f 6,7 t,1,1,f 6,7 t,f 7,6 t4 1, F 6,7 kt F 6,7 t, thus by lemma 2.1 r = s. This completes the proof of the theorem. Conclusion: Our theorem is an improvement of theorem 3.1 of saurabh manro [7]. In our theorem we do not require the completeness & continuity of the space and also condition (1) of [7, theorem 3.1]. Our theorem is true for any continuous t-norm. In our result we do not require to define many implicit relations. 4. References: B. Schweizer and A.Sklar, Probabilistic Metric Spaces, North Holland Series in Probability and Applied Math., 5(1983). G. Jungck, Commuting mappings and fixed point, Amer. Math. Monthly 83(1976), 261-263. G. Jungck and B.E. Rhoades, Fixed point theorems for occasionally weakly compatible mappings, Fixed PointTheory 7 (2006), 286-296. K. Menger, Statistical Metrices, Proc. Nat. Acad. Sci., U.S.A., 28(1942), 535-537. 41
M.A. Al-Thaga and N. Shahzad, Generalized I-nonexpansive selfmaps and invariant approximations, Acta Math. Sinica 24(5) (2008), 867-876. S. L. Singh and B.D. Pant, Common Fixed point theorems in probabilistic metric spaces and extension to uniform spaces, Honam Math. J. Phy., 6(1984), 1-12. S. Manro, A common fixed point theorem in menger space using property E. A. And implicit relation, J. of global res. in mathematical arch. 1(3), (2013), 40-44. V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory, 6(1972), 97-102. 42
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